The healing of the Centurion's servant (1-10) is an example of faith. Faith believes that what is said will be done. The Centurion believed that it was not necessary for Jesus to see the servant that was ill; all that was necessary was for Jesus to command that it would be done and the servant would be healed.
One commentator says that the saddest phrase in the gospel is in verse 12: "he was his mother's only son, and she was a widow;". In the economy of the time a woman was considered part of her father's house until she married of her husband's house until he died and then, usually, became resident with her eldest son until she died. A widow whose only son had died would be absolutely destitute in that time and place. Jesus not only restores the man who died back to life he has restored the widow's life as well.
The messengers from John the Baptist (18-35) raises some curious questions. Didn't John acknowledge that Jesus was the Messiah at Jesus' baptism? Doesn't he call Jesus "the Lamb of God" in John 1? How can John have been so sure a year ago and yet so full of doubt now? The answer lies in his situation. John is in prison. He has been put in prison because he has challenged the power of the day. What he challenged was right to challenge but when we challenge the powers of this world there are often consequences. John, the man of the desert, is locked up in a hole. I suspect that his imprisonment has worked on his faith and his hope and his understanding. Jesus demonstrates that he is the messiah by healing the sick, giving sight to the blind, preaching good news to the poor, etc. John was a great prophet in the tradition of the Old Testament prophets.
Jesus plays with the crowd (31-36). John the Baptist came wearing camel hair and eating bugs and honey and the religious establishment said "he has a demon (we might say he is crazy!)". Jesus comes eating and drinking and attending parties and human gatherings and the religious establishment says "he is a drunk and is clearly hanging out with the wrong crowd!" Jesus says they are like children playing in the marketplace. They sang the dirge (music of the funeral) and no one wanted to play funeral; they played the flute and no one wanted to play wedding. All of which is to say that the religious establishment is not interested in God's truth (which can come from the fasting in the wilderness John the Baptist types AND from the convivial hanging out with tax collectors and sinners types).
The story of the "sinful woman" would take too long to unpack here. What matters is the phrase "the one who has been forgiven much loves much." When we understand the depth of God's mercy extended to us we respond with gratitude. If we have never come to that understanding or do not believe we are in need of God's mercy we are less likely to live out that expression in love. The Pharisee feels justified by his religion and so does not even treat Jesus with the minimal of hospitality expected for the time -- water for the feet, a kiss of greeting, anointing oil on head. The women, whom Jesus likely encountered in the city street before entering the Pharisee's home, has experienced the transforming love and mercy of God through Jesus -- she expresses her gratitude through expressions of generous and extravagant love.